Asymptotic Analyses for Atmospheric Flows and ... - FU Berlin, FB MI

Klein, R.: Atmosphere Asymptotics and Numerics – AUTHOR’S UPDATED PRINT 765 

ZAMM · Z. angew. Math. Mech. 80 (2000) 11-12,765–785 

Klein, R. 

Asymptotic Analyses for Atmospheric Flows and the Construction 

of Asymptotically Adaptive Numerical Methods 

Prandtl’s boundary layer theory may be considered one of the origins of systematic scale analysis and asymptotics 

in fluid mechanics. Due to the vast scale differences in atmospheric flows such analyses have a particularly strong 

tradition in theoretical meteorology. Simplified asymptotic limit equations,derived through scale analysis,yield 

a deep insight into the dynamics of the atmosphere. Due to limited capacities of even the fastest computers, 

the use of such simplified equations has traditionally been a necessary precondition for successful approaches to 

numerical weather forecasting and climate modelling. 

In the face of the continuing increase of available compute power there is now a strong tendency to relax as 

many simplifying scaling assumptions as possible and to go back to more complete and more complex balance 

equations in atmosphere flow computations. However,the simplified equations obtained through scaling analyses 

are generally associated with singular asymptotic limits of the full governing equations,and this has important 

consequences for the numerical integration of the latter. In these singular limit regimes dominant balances of a 

few terms in the governing equations lead to degeneracies and singular changes of the mathematical structure 

of the equations. Numerical models based on comprehensive equation systems must simultaneously represent 

these dominant balances and the subtle,but important,deviations from them. These requirements are partly in 

contradiction,and this can lead to severe restrictions of the accuracy and/or efficiency of numerical models. 

The present paper makes a case for a somewhat unconventional use of the results of scale analyses and multiple 

scales asymptotics. It demonstrates how,through the judicious implementation of asymptotic results,numerical 

discretizations of the full governing equations can be designed so that they operate with uniform accuracy and 

efficiency even when a singular limit regime is approached. 

Keywords: Multiple Scales Asymptotics,Atmospheric Flows,Numerical Methods,Asymptotic Adaptivity 

1 Introduction 

Prandtl’s boundary layer theory may be considered one of the origins of systematic scale analysis and asymptotics 

in fluid mechanics, (see, e.g., [34, 41, 36]). Due to the vast scale differences in atmospheric flows, such analyses 

have a particularly strong tradition in theoretical meteorology. Classical textbooks and research monographs (see, 

e.g., [17, 32, 44]) explain how scaling arguments, in combination with order-of-magnitude-estimates of observed 

data, may be used to deduce simplified equation systems that capture the essence of atmospheric flows, while

766 ZAMM · Z. angew. Math. Mech. 80 (2000) 11-12 

suppressing unnecessary details. These systems often allow a more comprehensive understanding of many observed 

phenomena, because they contain only those physical interactions that are truly relevant. 

These limit equations generally emerge because of a singularity of the governing equations that is associated 

with the particular flow regime considered. This aspect is important in the context of numerical modelling based 

on the full governing equations. Flows that are generally within or close to the regime of validity of the asymptotic 

limit equations will be governed essentially by the dominant balances revealed by the asymptotics. Yet, a flow 

solver designed for the full governing equations will generally not be able to properly represent these and, at the 

same time, capture the remaining higher order effects. A prominent example, which is also relevant to atmospheric 

flow modelling, is the numerical simulation of low Mach number flows using fully compressible flow solvers. It has 

been shown, e.g., in [42, 38, 18], that standard compressible flow solvers become inaccurate or even fail completely 

for Mach numbers smaller than about M = 10 −2 . 

In the remainder of this paper we will recount in section 2aversion of the full three-dimensional compressible 

flow equations for flow in an atmospheric layer on the rotating earth in non-dimensional form. In section 3 we discuss 

qualitatively three singular asymptotic limit regimes that are relevant for different atmospheric flow applications, 

namely the quasi-geostrophic regime for synoptic scale flows (≈ 1000km), and the pseudo-incompressible and 

anelastic regimes for mesoscale flows on scales less than or equal to the pressure scale height (i.e., ≤ 10km). 

Section 4then provides some of results from (multiple scales) asymptotics of atmospheric flows. 

The scalings 

for this analysis are chosen so as to match the relevant numerically resolved scales in modern numerical weather 

forecast systems or regional climate models. Various issues regarding the numerical modelling of such flows on the 

basis of the fully three-dimensional compressible flow equations will be explained in section 5. A strategy for the 

design of suitable numerical schemes, which has been adopted in the author’s earlier work on low Mach number 

flows, will be summarized briefly. Section 6 provides conclusions and an outlook to future activities. 

2 Non-dimensional compressible flow equations in a rotating frame 

The discussions in this paper will be based on the compressible non-homentropic three dimensional flow equations 

in a rotating frame of reference (see, e.g., [17, 32, 44]): 

Sr ρ t + ∇·(ρ⃗v) = 0, 

Sr ⃗v t + ⃗v ·∇⃗v + 1 Ω 

Ro ⃗ × ⃗v + 1 1 

M 2 ρ ∇p + 1 ⃗ Fr 2 k = D ⃗ v , 

Sr p t + ⃗v ·∇p + γp∇·⃗v = D p . 

(1) 

Let ρ ref ,v ref ,p ref denote characteristic values of density, flow velocity, and pressure, let l ref ,t ref be characteristic 

length and time scales for the considered flow, and let Ω = 1 

day 

be the earth rotation frequency, and g the modulus


of the gravitational acceleration. Then the characteristic non-dimensional numbers Fr, M, Ro, and Sr are defined 

and labelled as 

Fr = 

v ref 

√ 

glref 

Froude Number 

M = 

v ref 

√ 

pref /ρ ref 

Mach Number 

Ro = 

Sr = 

v ref 

2Ωl ref 

l ref /t ref 

v ref 

Rossby Number 

Strouhal Number 

Furthermore, in (1), ρ, ⃗v, p are the non-dimensional density, velocity and pressure, t denotes time, and Ω, ⃗ ⃗ k are 

unit vectors pointing in the directions of the earth rotation axis and of the gravitational acceleration, respectively. 

⃗D v and D p are associated with the effects of viscous (turbulent) friction, dissipation, heat conduction, and radiation 

on the balances of momentum and energy. D p may also include effects of latent heat release, even though in this 

case the equation system would have to be extended to include the transport of the water phases. γ = c p /c v is the 

ratio of the specific heat capacities at constant pressure and at constant volume, respectively. 

In meteorological terminology, the homogeneous parts of (1) describe merely the “dynamics” of the atmosphere. 

In order to account for the so-called subgrid scale effects, suitable expressions representing turbulent 

transport, radiation effects, moisture transport, evaporation, condensation, precipitation, and possibly many other 

processes must be added to the “dynamic” part of the model. Such expressions are introduced in numerical atmospheric 

flow models, because it is generally not possible to resolve the atmospheric dynamics down to the very 

smallest flow scales by any realistic computational grid. Therefore, the effects of “unresolved scales” on the resolved 

ones must be parameterized, and the according terms are termed “the physics” of the model. They are subsumed 

here in D ⃗ v and D p . 

In the following we concentrate on the dynamics only and we discuss various singular flow regimes that 

emerge when one or more of the non-dimensional characteristic numbers defined above become very small or very 

large. 

3 Examples of singular limit regimes in atmosphere flows 

3.1 Quasi-geostrophic flow 

One prominent example of a singular flow regime is the quasi-geostrophic limit, [17, 32, 44, 30], derived comprehensively 

by Pedlosky [32]. Following his exposition, we consider 

1. a shallow atmosphere, with a vertical extent h ref ≪ l ref ,


2. horizontal scales small compared with the earth radius, so the flow occurs approximately in a tangent plane, 

3. horizontal scales over which the vertical component of the earth rotation vector changes appreciably, so that 

⃗ k · ⃗ Ω=1+βy, where y is a cartesian coordinate pointing in the meridional direction, and β is a constant, 

4. small vertical velocities w so that w/v ref ≪ O(h ref /l ref ), 

5. low Mach, Froude, and Rossby numbers so that 

M → 0, Fr → 0, Ro → 0, 

with 

M M2 

=1, and → 0 as M → 0 , (2) 

Fr Ro 

6. and unsteady flows with Sr = O(1) as M, Fr, Ro → 0. 

At the leading order, the vertical momentum balance yields hydrostatics, and the horizontal momentum balance 

shows that the leading order pressure and density depend on the vertical coordinate z only. The Coriolis effect 

induces a perturbation of order 

δ = M2 

Ro ≪ 1 , (3) 

so that density and pressure may be expanded as 

p = p(z)+δ ρ(z) p ′ (⃗x, z, t)+o(δ), 

ρ = ρ(z) 

( 

) 

1+δρ ′ (⃗x, z, t)+o(δ) , 

where 

dp 

dz = −ρ = −θ(z)p 1 γ . (4) 

Here ⃗x denotes the horizontal coordinates, and θ(z) is the leading order potential temperature distribution which 

must be considered an input to the problem formulation. The velocity field is expanded as 

⃗v = ⃗u qg +Ro(⃗u (1) + w (1) ⃗ k)+o(Ro) , (5) 

where vectors ⃗u denote the horizontal velocity components and w ⃗ k the vertical one. 

Asymptotic expansions then yield a closed set of equations for the quasi-geostrophic horizontal velocity ⃗u qg , 

the leading pressure perturbation p ′ and the potential temperature perturbation θ ′ = −ρ ′ + ρ 

γp (z) p′ : 

Horizontal momentum balance 

⃗ k × ⃗uqg + ∇ || p ′ =0 (6) 

Vertical momentum balance / hydrostatics 

θ ′ = ∂p′ 

∂z 

Transport equation for the vertical vorticity ζ qg = ⃗ k · (∇×⃗u qg ) 

( )( 

∂ 

∂t + ⃗u qg ·∇ || ζ qg + βy + 1 ( )) 

∂ ρ 

ρ ∂z S θ′ = 1 ( 

∂ ρD 

′ ) 

p 

ρ ∂z γpS 

(7) 

(8)


In these equations D ′ p is the suitably scaled source term from the original pressure evolution equation in (1), S is 

the static stability parameter defined by 

S = 1 θ 

dθ 

dz , (9) 

and ∇ || 

abbreviates the horizontal components of the gradient operator. 

We notice that (6)–(9) is the first closed set of equations obtained in this asymptotic regime of low Mach, Froude, 

and Rossby numbers. While (6) and (7) are the leading order horizontal and vertical momentum equations the 

vorticity transport equation (8) is a result of higher order balances. Thus, flow simulations based on more comprehensive 

equation systems, such as the three-dimensional compressible flow equations, must use highly accurate 

numerical schemes that can reliably represent such higher order balances without explicitly extracting them through 

asymptotics. 

Furthermore, the fact that the velocity field is related to a scalar function p ′ as in (6) introduces a compatibility 

constraint for the horizontal velocity field. One cannot assign an arbitrary two-dimensional initial velocity field, 

but must ensure that ⃗u qg is divergence free to begin with. In addition, a meaningful computation must generally 

start with an almost balanced flow, so that unbalanced fast wave components do not pollute the solution. 

(Note, however, that Embid and Majda [13, 14] reveal a flow regime in which the long time average effects 

of fast gravity waves do not influence the underlying slow time scale balanced mean flow. Thus, the often cost 

intensive procedure of finding balanced initial data may sometimes be omitted.) 

3.2 The incompressible, pseudo-incompressible, and anelastic approximations 

High frequency acoustic waves are generally considered unimportant for the purposes of weather prediction or 

climate modelling. Here “high frequency” is meant to denote acoustic modes with wavelengths on the order of 10 

km or less and characteristic frequencies higher than 1 min −1 . A plausibility argument states that sizeable elastic 

perturbations cannot establish in the atmosphere, because acoustic waves rapidly redistribute the associated energy 

and lead to an equilibration void of any acoustic modes. This intuitive explanation may be quantified through an 

asymptotic analysis for vanishing Mach number,[36, 28], that has been backed up by rigorous justifications, e.g., in 

[11, 22, 37]. The resulting set of zero Mach number, variable density flow equations without gravity and rotation 

can be written as 

ρ t + ⃗v ·∇ρ = −ρ∇·⃗v, 

⃗v t + ⃗v ·∇⃗v + 1 ρ ∇p′ = ⃗ D v , 

(10) 

∇·⃗v = D p 

γp ∞ 

.


Here p ′ is a higher order perturbation pressure satisfying p = p+M 2 p ′ , with p ≡ const.. (Notice that for combustion 

in closed systems the background pressure p may be time dependent, but will always be spacially homogeneous.) 

The (elliptic) divergence constraint (10) 3 replaces the total energy or pressure evolution equation. It effectively 

suppresses acoustic modes and is therefore extremely important for numerical flow simulations: Fast acoustic 

wave propagation is eliminated and thus will neither lead to undesired time step restrictions, nor to unphysical 

oscillations that may occur when straight-forward discretizations are used to represent hyperbolic equation systems 

(see, e.g., [27]). The constraint does induce a complication, in that it establishes an elliptic, Poisson-type equation 

for the pressure perturbation p ′ , namely 

which 

∇·( 1 ρ ∇p′ )=−∇ · (⃗v ·∇⃗v − ⃗ D v ) − ∂ ∂t 

( ) 

Dp 

, (11) 

γp 

• is not trivially solved numerically, and 

• has no connection with the thermodynamic equation of state which for compressible flows would establish a 

relation between the density and pressure fields. 

In practice, these complications are often considered less severe than the time step constraints associated with a 

proper representation of anyway unimportant acoustic waves. 

Durran [10] describes two similar velocity divergence constraints that are used frequently in the formulation 

of numerical atmosphere flow models when the horizontal scales considered are comparable to the vertical ones 

(see also [17, 44, 12]). On such scales one may describe, e.g., the transport of pollutants in the atmosphere after 

they have exited a smoke stack (relevant length scales: ≈ 100 m – 1 km), or one may simulate the formation and 

evolution of convective clouds on scales comparable to the atmospheric scale height of about 10 km. 

The first constraint, 

∇·(ρ⃗v) =0 (12) 

leads to the “anelastic approximation”, [3, 31, 9]. The second yields the “pseudo-incompressible approximation”, 

[9], and reads 

∇·(ρθ⃗v) = 

D p 

γp γ−1 

γ 

. (13) 

Here 

θ = p1/γ 

ρ 

(14) 

is the standard “potential temperature” except for a constant factor. The potential temperature is a function of 

the thermodynamic entropy, (see, e.g., [17], section 3.7.3). The overbar denotes horizontal averaging at constant 

height, z = const..


If either one of these constraints is adopted in a numerical model, the desired effect of suppression of acoustic 

modes is obtained. Yet, these constraints are not equivalent and which one best suits a given situation must be 

carefully evaluated. Estimates of the respective regimes of validity of the constraints in (12) and (13) as well as a 

discussion of consequences for numerical discretizations will be given in section 4.1 below. Here we merely re-iterate 

an interesting observation from [44, 5]: 

If the anelastic and the pseudo-incompressible approximations hold simultaneously, then either 

• the vertical flow velocity w satisfies the “diagnostic relation” 

or 

w = D ( )−1 

p 1 ∂θ 

, (15) 

γp θ ∂z 

• the potential temperature has an expansion θ = θ ∞ +M 2 θ (2) (⃗x, z, t), where θ ∞ ≡ const. 

Notice that the second option has been assumed by Ogura and Phillips [31] in their derivation of the anelastic 

approximation. (See also sections 45, 46 of [44].) 

4 Asymptotic analysis 

4.1 The anelastic and pseudo-incompressible divergence constraints 

Here we consider atmospheric motions on length scales of up to the pressure scale height, i.e., 

l ref ≤ h sc ≈ 10km , 

and we assume an isotropic scaling of characteristic lengths, i.e., no shallowness is assumed. Also we are interested 

in convective time scales, so that we choose 

t ref = l ref 

v ref 

and Sr ≡ 1 . 

The Rossby number may be estimated as 

Ro = 

v ref 

> 

v ref 

≈ 5 , 

2Ωl ref 2Ωh sc 

so that 1/Ro ≤ O(1) and the Coriolis effects are non-singular. 

To assess the relative order of magnitude of the Mach and Froude numbers we first observe that the Froude 

number based on the pressure scale height actually equals the Mach number. The pressure scale height may be 

assessed using the approximate hydrostatic balance ∂p/∂z ≈−ρg, and the scale height is defined as the vertical


distance over which the pressure changes appreciably, i.e., by an amount comparable to the reference pressure at 

the ground. Therefore 

∣ ∣∣∣ ∂p 

h sc ∂z∣ ≈ h scρ ref g := p ref , or h sc g ≈ p ref 

. 

ρ ref 

Thus, from the definitions of the characteristic numbers we have 

Fr ≥ M (with Fr = M for l ref = h sc ) . 

The pseudo-incompressible approximation. 

It is generally observed that pressures in the troposphere 

at any given location do not change appreciably in comparison with the background pressure of about 1 

bar. Thus pressure variations are always small and we may write 

p = p(⃗x, z)+O(ε p ) where ε p ≪ 1 , (16) 

and p(⃗x, z) is a time independent local mean pressure. Also, high frequency pressure fluctuations on time scales 

much shorter than the convective time scales are either absent or of such low amplitudes that they do not appreciably 

affect the pressure time derivative. As a consequence we have 

∂p 

∂t = O(ε p,t) ≪ 1 . (17) 

With these estimates, the pressure evolution equation (1) 3 yields 

⃗v ·∇p + γp∇·⃗v = γp γ−1 

γ 

∇·(p 1 γ ⃗v) = Dp + O(ε p,t ) . (18) 

If the pressure perturbations of order O(ε p ) allow the additional estimate 

∇p = ∇p + O(ε p,x ) with ε p,x ≪ 1 , (19) 

we may neglect errors of order O(ε p ,ε p,t ,ε p,x ) to find the “generalized pseudo-incompressible constraint” 

∇·(p 1 γ ⃗v) = 

D p 

γp γ−1 

γ 

. (20) 

The only approximations that are needed to obtain this result concern the smallness of the overall pressure variations 

and of the pressure time and space derivatives. 

Thus this divergence constraint is a direct consequence of a 

degeneration of the pressure evolution, and not one of mass conservation (see also [23]). In contrast, the original 

pseudo-incompressible divergence constraint from (13) necessitates the additional requirement ρθ = p 1 γ . Unless 

one defines θ and ρ through 

θ = 〈θ〉 and ρ = 〈 1 ρ 〉−1 (21) 

with a suitable averaging operator〈·〉, this requirement will be satisfied only when fluctuations of potential temperature 

and density are small. This would not be the case, e.g., for hot exhaust gases exiting a smoke stack.


The low Mach number limit for small scales. When l ref ≪ h sc , we have Fr ≫ M and an 

appropriate asymptotic pressure expansion reads 

p = p + ε Fr p (Fr) (⃗x, z, t)+M 2 p (2) (⃗x, z, t)+o(ε Fr , M 2 ) (22) 

where 

ε Fr = M2 

Fr 2 ≪ 1 . (23) 

The momentum equation at order O(M −2 ) reads 

∇p ≡ 0 , (24) 

so that p is a constant. For this case the generalized pseudo-incompressible constraint from (20) reduces to the 

standard divergence condition for non-adiabatic zero Mach number flow 

∇·⃗v = D p 

γp . (25) 

This constraint is common, e.g., in the theory of low Mach number combustion, [28]. 

When Fr = O(1) we have ε Fr =M 2 and the pressure perturbations p (Fr) and p (2) need not be distinguished. 

In this regime, which describes flow fields on the 10 m scale, the leading order system of governing equations is that 

presented in (10), provided the gravitation term is included in the momentum source ⃗ D v . There are interesting 

intermediate regimes for which the condition in (23) holds, and for which there is a non-trivial distinguished limit 

Fr = M α with 0


The horizontal momentum balance at order O(M −2 ) yields 

∇ || p ≡ 0 or p = p(z) . (30) 

Here ∇ || abbreviates the horizontal components of the gradient operator. From (29) and the (assumed) time 

independence of p we conclude that 

ρ (0) ≡ ρ(z) . (31) 

With this result we obtain the anelastic divergence constraint (12) as the leading order continuity equation, i.e., 

∇·(ρ⃗v (0) )=0. (32) 

We realize furthermore that all the conditions needed to justify the pseudo-incompressible divergence constraint 

from (20), namely p = p(⃗x, z)+ε p , ∂p/∂t = O(ε p,t ), and ∇p = ∇p + O(ε p,x ) are satisfied, with ε p = ε p,t = 

ε p,x =M 2 . Thus, the anelastic and the pseudo-incompressible divergence constraints hold simultaneously! Using 

(29), (30), and (31) we conclude, after some straight-forward manipulations, that 

( 

) 

w (0) 1 dp 1 γ 

dz − 1 dρ 

= w (0) 1 dθ 

ρ dz θ dz = D(0) p 

γp , (33) 

p 1 γ 

where w (0) is the leading order vertical velocity and θ = p 1 γ /ρ. 

At this point we have to distinguish different 

regimes concerning the sign and order of magnitude of the dry adiabatic stability parameter 

1 dθ 

θ dz = S =: M2 N 2 , (34) 

where N is the — non-dimensional — “buoyancy frequency” or “Brunt-Väisälä” frequency. This quantity is a 

measure of the oscillation frequency of particles that are vertically displaced in a stably stratified atmosphere with 

dθ/dz > 0, (see, e.g., [17, 32, 44, 12]). For many, if not most, practical purposes one may assume stable stratification 

with N 2 > 0 in the troposphere and stratosphere. Pedlosky [32] provides an order of magnitude estimate (see his 

eq. (6.4.13)), which in the present notation amounts to 

M 2 N 2 = 1 θ 

dθ 

dz = O(10−1 ) . (35) 

With M ≈ 0.03, v ref = 10m/s, l ref = h sc =10 4 m,t ref = l ref /v ref this amounts to a dimensional frequency of 

N/t ref =10 −2 s −1 , (see also [32], Fig. 6.4.2, and [17], page 52). 

Stable Stratification: Assuming now that the stability parameter is non-zero and positive, the simultaneous 

realization of the anelastic and pseudo-incompressible divergence constraints from (33) leads us to a diagnostic, 

algebraic relation for the leading order vertical velocity, 

w (0) = D(0) p 

γp 

( 1 

θ 

)−1 

dθ 

dz 

if 

dθ 

dz > 0 , (36)


as pointed out in section 1. Here is a summary of the relevant asymptotic limit equations for the horizontal velocity 

⃗u, the perturbation pressure p (2) , and the vertical velocity w in this regime. 

⃗u t + ⃗u ·∇ || ⃗u + w⃗u z + 1 ρ ∇ || p (2) = ⃗ D u , 

∇ || · ⃗u = − 1 ∂ρw 

ρ ∂z , 

( )−1 

w = D(0) p 1 dθ 

. 

γp θ dz 

(37) 

The energy source term D p may depend on w, ⃗u, ρ, p, and on additional variables, such as humidity, for which 

one would have to include additional inhomogeneous scalar transport equations. Since the purpose of the present 

derivations is to point out the structure of the low Mach / low Froude number singularities rather than being 

comprehensive, we omit detailled specifications of D p here. 

The equations in (37) are strongly degenerate, [44], in that the vertical momentum balance is no longer part 

of the prognostic set of evolution equations. In this regime, vertical motion can take place only if there is sufficiently 

strong heat input or heat loss through sources such as latent heat release of absorption of radiation. In the absence 

of heat exchange there is no vertical motion and the airflow occurs in vertically stacked, decoupled layers. One 

consequence of the algebraic constraint in (36), which is consistent with the discussions in [39], is that it generally 

forces airflow around topographical obstacles rather than allowing flow over them. 

Nearly Neutral Stratification: The desire to model events of “deep convection” in well-mixed atmospheric 

boundary layers has led Ogura and Phillips [31] to introduce the assumption of near neutral stability with 

1 dθ 

θ dz = O(M2 ) . (38) 

With the dimensional parameters as in the previous paragraph, this corresponds to N/t ref ≈ 10 −3 s −1 , and thus 

to a much weaker stratification than before. Zeytounian ([44], chapter 45) provides a compact derivation of the 

relevant governing equations for the full three-dimensional velocity field, ⃗v, and the second order perturbation of 

the potential temperature, θ (2) . In our notation these equations read 

⃗v t + ⃗v ·∇⃗v + 1 ρ ∇p(2) = ⃗ D v + θ (2) ⃗g, 

θ (2) 

∇·(ρ⃗v) = 0, 

(39) 

t + ⃗v ·∇θ (2) = D(2) p 

γp . 

Importantly, in this regime the energy input represented by the source term D p must be small of order O(M 2 ), i.e., 

D p =M 2 D (2) 

p + o(M 2 ) . (40) 

The leading order pressure and density stratifications now correspond to a neutrally stable atmosphere and are


explicitly given by 

p(z) = 

( 

1 − γ − 1 

γ 

) γ 

z 

γ−1 

, 

1 

ρ(z) =p(z) 

γ . (41) 

Zeytounian [44] discusses two-dimensional steady state solutions of the above equations. He reduces the equations 

to a single second order elliptic equation for a stream function ψ. This special case provides an excellent basis 

for the design of test problems for comprehensive solvers of the three dimensional anelastic weak stratification 

equations from (39). 

4.2 Long wave length dynamics on short convective time scales 

Thus far we have considered asymptotic flow regimes that were characterized by single length and time scales. 

We did account for shallow atmospheres with vertical extent small compared to the horizontal scales, but in each 

direction only a single characteristic scale has been accounted for. Furthermore, the reference time scale has been 

the characteristic time scale of convection, i.e., t ref 

= l ref /v ref . Meteorological applications, however, generally 

feature a multitude of scales. 

On the one hand, there are the “unresolved” or “subgrid scales”, which are smaller than what can be 

resolved by the computational grids of atmospheric flow models. The overall effects of these subgrid scales on the 

resolved ones is a very subtle and important issue. The complexity of the problem is even higher than that of 

classical turbulence modelling. Many of the simplifying assumptions that are often used in this latter area, such 

as homogeneous isotropic turbulence characteristics, are not valid in the atmosphere due to anisotropic scalings, 

the influence of gravity and stratification, etc.. In addition, there are microscale processes, e.g., in the context of 

cloud formation, that can only be described by multi-phase fluid dynamics. We will not address the problems of 

“parametrization of subgrid scale effects” here. 

Another multiple scales aspect arises from the continuous increase of modern computational capacities. It 

is now possible to simultaneously resolve a multitude of length scales within a single computation. For example, 

the current generation of weather forecast codes uses a horizontal grid spacing as small as ∼ 3km, while spanning 

overall horizontal distances of thousands of kilometers. Some of the consequences of the simultaneous presence of 

multiple length scales can be explored using asymptotic multiple scales expansions, and this is the topic of the 

present section. The following discussions will be based on the detailed derivations in [5]. 

During previous discussions we have seen already that the mathematical structure of the effective dynamical 

equations changes drastically with the considered length scales. When considering multiple length scales in a single 

solution, a subtle issue regarding the underlying characteristic time scales arises in addition: At low flow Mach 

numbers the characteristic time scale for convection is always much longer than that of acoustic wave propagation


for a given length scale. If, however, the largest present scale, l max , exceeds the smallest one, l min , by a factor 

of 1/M or more, then the characteristic time of convection on the l min scale becomes comparable to the acoustic 

time scale for l max . 

It is then not at all clear that the “filtering of acoustic modes”, as implied, e.g., in the 

quasi-geostrophic, anelastic, and pseudo-incompressible approximations, should affect all flow scales, or whether 

it should be restricted to the smallest scales only. In fact, there are very long wavelength combined acoustic — 

gravitational wave modes, the so-called Lamb waves (see [17], p. 504–506), which are inherently compressible and 

must be accounted for together with the quasi-incompressible small scale dynamics. 

Botta et al. [5] consider flows with a smallest scale comparable to the pressure scale height h sc and chose 

reference length and times for non-dimensionalization according to 

l ref = h sc ≈ 10 4 m , 

t ref = l ref 

v ref 

≈ 10 3 s (42) 

(i.e., the reference flow velocity is v ref ≈ 10m/s). However, they are interested at the same time in much larger 

horizontal scales 

l ac = 1 M l ref ≈ 300km (43) 

for which one finds the acoustic time scale to match the convection time scale for l ref 

l ac 

= Ml ac 

= l ref 

= t ref . (44) 

c ref v ref v ref 

As discussed earlier, the Froude and Rossby numbers based on the reference scales may be assessed through the 

distinguished limits 

Ro = O(1) , 

Fr 

= O(1) as M → 0 . (45) 

M 

Notice that the effective Rossby number for the larger “acoustic” scales becomes 

Ro ac = v ref 

Ωl ac 

= O(M) as M → 0 , (46) 

so that we may expect the effects of rotation to achieve increasing importance at the larger length scales. 

Following the low Mach number single time – multiple space scale analysis for weakly compressible flows from 

[23], Botta et al. then choose an asymptotic ansatz for solutions of (1) of the form 

U(⃗x, z, t;M)=U (0) (⃗x, M⃗x, z, t)+MU (1) (⃗x, M⃗x, z, t)+M 2 U (2) (⃗x, M⃗x, z, t)+o(M 2 ) , (47) 

where U =(ρ, p, ⃗u, w) t denotes the solution vector. Without going into details we will summarize here some of the 

key results of the subsequent analysis. For future reference we note that horizontal spatial gradients for the above 

solution ansatz translate as follows 

∇ || U = ∑ ν 

M ν (∇ ⃗x +M∇ ⃗ξ ) U ν , (48) 

where ⃗ ξ =M⃗x.


Small scale dynamics: 

On the small scales comparable to h sc it is found that 

∇ ⃗x p (0) = ∇ ⃗ξ p (0) = ∇ ⃗x p (1) ≡ 0 so that p (0) = p(z,t) , p (1) = p (1) ( ⃗ ξ,z,t) . (49) 

Consistent with observations, temporal variations of the background pressure p are suppressed, so that p = p(z). 

Next one recovers the pseudo-incompressible flow equations for stratified flows from (37), except for one 

modification concerning the influence of long wavelength pressure gradients. The horizontal momentum equation 

now reads 

⃗u t + ⃗u ·∇ ⃗x ⃗u + w⃗u z + 1 ρ ∇ ⃗x p (2) = ⃗ D Ro 

u − 1 ρ ∇ ⃗ ξ 

p (1) , (50) 

where we have included the Coriolis terms in ⃗ D Ro 

u 

to abbreviate the notation. One may expect to find the fully 

three-dimensional anelastic approximation from (39), plus the long wave pressure gradient term as in (50), if 

variations of the potential temperature are assumed to be of order O(M 2 ) only. This case was not discussed in [5]. 

The only new effect on the small scales is thus the pressure gradient term 1 ρ ∇ ξ ⃗ p (1) , which is responsible for 

a bulk acceleration of the fluid. Notice that the ⃗x-dependence of the second order pressure p (2) is determined by a 

small scale divergence constraint analogous to (37) 2 . 

Large scale dynamics: 

In a standard fashion, the relevant equation for the dynamics on the 1 M l refscale 

are obtained by horizontally averaging over the small scale coordinate ⃗x. The ⃗x-averaging operator will be 

abbreviated by 〈·〉. The large scale dynamics involves three variables, namely the long wave pressure mode p (1) , 

the averaged vertical component of vorticity ζ = ⃗ k ·∇ ⃗ξ 

×〈⃗u (0) 〉, and the averaged first order vertical velocity 

w (1) = 〈w (1) 〉. The long wavelength dynamical equations for the considered regime read 

( ∂2 

∂t 2 − c2 ∇ ⃗ξ 2 ) p (1) = (1− c 2 ∂ ∂z ) ∂ ∂t (ρ w(1) ) − c 2 ζ + Q p (1) , 

N 2 c 2 (ρ w (1) ) = −(1 + c 2 ∂ ∂z ) ∂ ∂t 

p (1) + Q ρw (1) , 

(51) 

∂ 

∂t ζ = 1 

c 2 ∂ 

∂t p(1) + ∂ ∂z (ρ w(1) ) + Q ζ 

. 

Here Q i are inhomogeneous terms that combine the effects of small scale averages of nonlinear terms, and of the 

momentum and energy inhomogeneities ⃗ D v ,D p from (1). The system in (51) supports internal gravity waves, long 

wave acoustics, and the Lamb wave. It represents the link between the multiple length scale asymptotics from [5] 

and classical analyses for small perturbations from a state of rest. 

We notice also that the in the stationary limit (∂/∂t ≡ 0) the first equation in (51) reduces to the pressure 

Poisson equation from the quasi-geostrophic theory obtained by applying the horizontal divergence ∇ || 

· (·) to 

the geostrophic balance equation (6). Thus we verify that the Coriolis effects achieve dominant importance for 

quasi-steady flow on the larger ⃗ ξ-scale as announced earlier.


5 Numerical issues 

5.1 Discussion of the singular limit regimes 

In the previous sections we have provided a summary of some important singular limit regimes for atmospheric 

flows. Simplified asymptotic limit equations have been discussed that can be derived systematically from the fully 

three-dimensional compressible flow equations by considering various distinguished limits connecting the Mach, 

Froude, and Rossby numbers. (We have focused on convective time scales, so that the Strouhal number has been 

set to unity throughout). Based on these considerations we will now point out a number of difficulties that must be 

addressed when near-singular solutions to the original compressible flow equations are to be obtained via numerical 

integration. 

Low Mach number small scale flows: 

Consider first the regimes of pseudo-incompressible and 

anelastic flows from section 3.2 and 4.1. It has been pointed out that the stratification of potential temperature 

plays an crucial role for the flow dynamics. Under near neutral conditions, where θ = θ ∞ +M 2 θ (2) , there is fully 

three-dimensional dynamics, whereas the vertical motion is strongly blocked by buoyancy forces if the stratification 

is stronger. Furthermore, in this regime there is an asymptotic decoupling between the thermodynamic component 

of the pressure, the background pressure p, and the pressure gradients in the momentum equation, which are 

represented by the second order pressure p (2) . 

Suppose now that a fully compressible flow solver using pressure, density, and velocity as the dependent 

variables were used to compute a weakly stratified flow. Let us suppose further that a second order accurate discretization 

is used and that the numerical resolution is of the order of h = l ref /∆x ≈ 10 −2 , where ∆x characterizes 

the mesh size of the computational grid. Local truncation errors within the pressure and density evolution equations 

will then be of order δρ, δp = O(h 2 ) ∼ 10 −4 . After a limited number of time steps the potential temperature 

θ = p 1 γ /ρ will have accumulated errors of order O(10 −3 ), which is comparable to the square of the Mach number. 

Thus, unless additional special measures are taken, the buoyancy forces in the vertical momentum equation will 

be dominated by the net effect of truncation errors, and an accurate prediction of vertical velocities cannot be 

expected. 

Suppose next that the same flow solver is used to compute a flow field with stable stratification as described 

by equations (37). The diagnostic constraint for the leading order vertical velocity states that, upon heat addition, 

a mass element will move vertically within the stratification so as to maintain a position of neutral buoyancy. This 

motion is a result of an intimate coupling that simultaneously involves the pressure, density, and vertical momentum 

equations. Unless specially designed to respect this balance, a fully compressible flow solver will respond to heat 

addition by generating unphysical acoustic waves with amplitudes of the order of the numerical discretization error.


Buoyancy imbalances induced by truncation errors can in addition excite unphysical internal gravity waves. 

Another deterioration of accuracy with decreasing Mach number is associated with the asymptotic pressure 

decomposition. Compressible flow solvers which feature a single pressure variable without explicitly implementing 

the asymptotic scalings have been observed to fail for sufficiently small Mach number in [6, 42, 38, 18]. This failure 

can be traced to back to cancellation of significant digits in the discretization of the pressure gradient. There is a 

large volume of literature addressing the issue of low Mach number flow numerics. Besides the references already 

cited here is an additional, unfortunately still incomplete, list for further reading, [1, 2, 4, 6, 16, 21, 23, 25, 26, 29, 40]. 

(See also section 5.2 below.) 

In addition to these accuracy issues, compressible flow solvers often become inefficient at low Mach numbers, 

because they are subject to stability related time step restrictions associated with the acoustic wave speed. The 

regime of anelastic, weakly stratified flow, (39), allows the construction of more efficient schemes with time steps 

restricted by the magnitude of the convection velocity rather than the sound speed. The complexity of an associated 

computational model is comparable to that of a three-dimensional incompressible flow solver. If one is interested 

furthermore in flows with adiabatically stable stratification, (37), then the vertical velocity is given diagnostically 

and one is left with two-dimensional divergence constraints for the horizontal velocity in each horizontal layer of the 

computational grid. Imposing a set of independent horizontal divergence constraints should be much more efficient 

than a single three-dimensional one, and a considerable further gain in computational speed can potentially be 

achieved. 

Multiple scales: 

Large (synoptic) scale flows in the atmosphere may be approximated quite well 

by the quasi-geostrophic theory summarized in section 3.1. However, the quasi-geostrophic equations are highly 

filtered in that they suppress most of the wave phenomena in the atmosphere, such as long wave acoustic gravity 

waves and internal gravity waves. Furthermore, modern computational capacities allow the simultaneous numerical 

representation of such long wavelength phenomena and flow features on length scales of a few kilometres. We have 

seen that the anelastic and pseudo-incompressible flow limits, which are relevant for these smaller scales, are 

described by equation systems of a very different nature. It is a highly non-trivial challenge to construct numerical 

schemes which represent the almost balanced large scale flows, at the same time resolve the much richer small 

scale dynamics, and provide high accuracy on both. In the next section we briefly summarize a strategy for the 

construction of multiple scales numerics that combines ideas from asymptotic analysis with numerical multi-grid 

techniques.


5.2 Asymptotically adaptive numerical methods 

Here we consider low Mach number flows without gravity and rotation that are characterized by a single time scale, 

but multiple space scale analogous to what has been presented in section 4.2. In [23] the author derives simplified 

asymptotic limit equations that describe interaction of the long wave acoustic and short wave quasi-incompressible 

flow phenomena. Using a solution ansatz analogous to that in (47) the author obtains the following leading order 

set of equations for inviscid, adiabatic flows: 

Small scale quasi-incompressible flow 

ρ (0) 

t + ∇ ⃗x · (ρ (0) ⃗v (0) ) = 0 

(ρ (0) ⃗v (0) ) t + ∇ ⃗x · (ρ (0) ⃗v (0) ◦ ⃗v (0) )+∇ ⃗x p (2) = −∇ ⃗ξ p (1) 

(52) 

∇ ⃗x · ⃗v (0) = 0 

Long wave acoustic modes 

ρ (0) t = 0 

(ρ (0) ⃗v (0) ) t 

+ ∇ ⃗ξ p (1) = 0 

(53) 

p (1) 

t + γP 0 ∇ ⃗ξ · ⃗v (0) = 0 

Here P 0 is the leading order pressure, which is constant in this regime. As in section (47), the first order pressure 

does not have small scale variation, so that ∇ ⃗x p (1) ≡ 0 and p (1) = p (1) ( ξ,t). ⃗ 

Solution techniques for the full compressible flow equations from (1) are suggested in [15, 16, 23, 24, 33] 

which are designed to operate efficiently and accurately in the present asymptotic regime of weakly compressible 

low Mach number flows. The idea is to first obtain efficient and accurate estimates of the key physical effects in 

the flow by using appropriate, optimized numerical methods to integrate the asymptotic limit equations over one 

time step. Then, ideas similar to those behind projection methods for incompressible flows, [19, 7, 8], are used to 

compose a final solution that actually obeys the full governing equations instead of merely the asymptotic limit 

equations. Here is a qualitative description of the key steps need to address the flow regime described by (52), 

(53). 

In step 1 one obtains an explicit estimate of the effects of nonlinear convection using an upwind based higher 

order discretization. The time step restriction for this procedure is given by a stability constraint based on the 

maximum flow velocity (instead of on the sound speed). 

Step 2 consists of (i) a filtering technique that extracts the current long wave components of the pressure, 

momentum, and velocity fields, p (1) , ⃗v, ρ⃗v. These data are then used to solve integrate (53) over one time step


using a discretization that overcomes the sound speed based stability constraints. Here one may use an implicit 

discretization on the original mesh. Alternatively, one may consider the long wave – short wave decomposition as 

the basis of a physics-induced multi-grid technique and solve (53) by an explicit method on a coarse grid, [33]. 

The final step determines the small scale, second order pressure p (2) , which is responsible for guaranteeing 

that the discrete flow field locally obeys a divergence constraint as in (52) 3 . This step corresponds to a “projection 

step” in classical incompressible flow solvers. 

Notice that steps 2 and 3 yield independent evaluations of discrete analogues of the first and second order 

pressures p (1) ,p (2) from the asymptotic analysis. In the numerical scheme, these pressure contributions are computed 

on the basis discretizations that are non-singular as the Mach number vanishes. Thus, the above-mentioned 

cancellation of significant digits is avoided. Only after completion of the time step are the pressure contributions 

re-composed to yield the full thermodynamic pressure. 

A subtle issue in this approach is the fact that the numerical discretization makes explicit use of the current 

value of the Mach number. It is thus necessary to implement intelligent filtering techniques that extract the long 

wave and short wave solution components and, simultaneously, simultaneously yield the instantaneous relevant 

value of the Mach number. This issue has been addressed in [24]. 

Depending on the flow regimes that a computational code is mostly used for, the detailed components for 

steps 1 to 3 may vary. The original idea in [23] was to extend a higher order shock capturing technique for fully 

compressible flows to the regime of small and even zero Mach number, and to include long wave acoustic effects. 

The desired extension to zero Mach number, variable density, multi-dimensional flows has been achieved in [35], 

code versions for flows including long wave acoustics have been presented in [15, 16]. 

If one is interested mostly in flows at very low Mach number, then the overhead associated with full-fledged 

shock capturing techniques are unnecessary and simplifications can be introduced. Thus, Roller et al. [33] start from 

SIMPLE-type incompressible flow solvers, (see, e.g., [21]), and extend these technologies to the weakly compressible 

flow regime using the above techniques. Another alternative strategy has been proposed in by Worlikar et al. [43] 

in order to simulate the flow in a thermo-acoustic device. Their scheme is based on a vorticity stream function 

formulation for the small scale quasi-incompressible flow, and uses the modified Godunov-Type scheme from [23] 

for long wave acoustics. 

6 Conclusions and outlook 

The “traditional” aim of scale analysis in meteorology is to obtain simplified asymptotic limit equations that are 

easier to solve than the more comprehensive fully compressible flow equations. There is a large number of such 

simplified model equations, and each has its particular advantages and shortcomings in practice. In recent years the


rapid increase of compute power has stimulated strong efforts to relax the various simplifying scaling assumptions 

associated with asymptotic limit equations and to tackle the numerical solution of the full governing equations 

directly. It has become clear, however, that a lack of computing power has not been the only obstacle in attempts 

at solving these equations in earlier decades. Additional numerical issues arise, which are intimately related to 

the fact that scale analysis and asymptotic modelling have been successful in the first place. The derivation of 

simplified asymptotic models usually takes advantage of a singular degeneration of the governing equations in the 

respective limit regime, and it is these singularities that can induce severe numerical stiffnesses, and dynamic range 

problems. 

This paper advocates a so far relatively unconventional use of asymptotic scale analysis in the context of 

atmosphere flow modelling (see, however, [20]). We propose to construct new classes of “asymptotically adaptive 

numerical methods”, which do solve the full three dimensional compressible flow equations, but use the results 

of asymptotic scale analysis in the design of the discretizations. 

Such a scheme would assess a small number 

of nondimensional characteristic numbers “on the fly” during a computation. These characteristic numbers are 

chosen so as to indicate whether the current flow state is or is not within the vicinity of a singular limit regime. 

As a singular limit is approached, the discretizations automatically adapt and they merge into a scheme for the 

asymptotic limit equations when the limit is actually achieved. 

The strategy has been explained for the case of low Mach number multi-dimensional flows, for which it has 

been shown to be successful in a series of publications. 

Acknowledgement. The author gratefully acknowledges the continous constructive co-operation with Dr. 

Ann Almgren,Lawrence Berkeley National Laboratory,USA,Dr. Omar M. Knio,Johns Hopkins University,Baltimore, 

MD,USA,and Dr. Nicola Botta,Potsdam Institute for Climate Impact Research,Germany. This work has been sponsored 

partly by the Deutsche Forschungsgemeinschaft under Grant KL 611/6-1,2. 

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Address: Prof. Rupert Klein,Potsdam Institut für Klimafolgenforschung,Telegrafenberg C4,14412 Potsdam,Germany, 

email: rupert.klein@pik-potsdam.de

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